Algebra Universalis最新文献

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, (texttt{DNA})-logics and dependence logic.

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Słupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4 have this property.

We develop a common semantic framework for the interpretation both of ({textbf {IPC}}), the intuitionistic propositional calculus, and of logics weaker than ({textbf {IPC}}) (substructural and subintuitionistic logics). To this end, we prove a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the full subcategory of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.

From work of Vaughan-Lee in [12] it follows that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan-Lee gives an explicit example of a nilpotent loop of order 12 that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.

As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity. In particular, it is shown that an algebraic closure system with arity n in which the empty set is closed and every finitely generated closed set is countable can be realized by a single ((n+1))-ary operation. The algebraic closure system of cosets on any group is realized by the single ternary Mal’cev term (xy^{-1}z). It is shown that the closure system of cosets on an Abelian group A can be realized by a single binary operation if and only if A has at most one element of order 2. Similar results are obtained for modules over an arbitrary ring.

Let ({mathcal {V}}) be a variety of algebras of some type (Omega ). An interest to describing automorphisms of the category (Theta ^0 ({mathcal {V}})) of finitely generated free ({mathcal {V}})-algebras was inspired by development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and applied by B. Plotkin and the author. The method is to find all terms in the language of a given variety which determine such (Omega )-algebras that are isomorphic to a given (Theta ^0 ({mathcal {V}}))-algebra and have the same underlying set with it. But this method can be applied only to automorphisms which take all objects to isomorphic ones. The aim of the present paper is to suggest another method which works in more general setting. This method is based on two main theorems. The first of them gives a general description of automorphisms of categories which are supplied with a faithful representative functor into the category of sets. The second one shows how to obtain the full description of automorphisms of the category (Theta ^0 ({mathcal {V}})). This part of the paper ends with two examples. The first of them shows the preference of our method in a known situation (the variety of all semigroups) and the second one demonstrates obtaining new results (the variety of all modules over arbitrary ring with unit). The last section contains some applications to universal algebraic geometry.

We introduce the category of Heyting frames, those coherent frames L in which the compact elements form a Heyting subalgebra of L, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.

The collection of all cohereditary classes of modules over a ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are the conatural classes of R-modules. In this paper we extend some results about cohereditary classes in R-Mod to the category (mathcal {L_{M}}) of linear modular lattices, which has as objects all complete modular lattices and as morphisms all linear morphisms. We introduce the big lattice of conatural classes in (mathcal {L_{M}}), and we obtain some results about it, paralleling the case of R-Mod and arriving at its being boolean. Finally, we prove some closure properties of conatural classes in (mathcal {L_{M}}).

In the paper, a basis of identities for the variety generated by the class of groupoids that are generalized subreducts of Tarski’s algebra of relations is found. It is also proved that the corresponding class of groupoids does not form a variety.

We investigate characterizations of the Galois connection ({{,textrm{Aut},}})-({{,textrm{sInv},}}) between sets of finitary relations on a base set A and their automorphisms. In particular, for (A=omega _1), we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under ({textrm{sInv Aut}}). Our structure (A, R) has an (omega )-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.